Japan “Sangaku” Research Institute

Problem Let the midpoints of sides $BC, \ CA$ and $AB$ of a triangle $ABC$ be $D, \ E$ and $F$, respectively. Also, let $G$ and $H$ be the feet of perpendiculars drawn from $B$ and $C$ to any line passing through $A$, respectively, and $I$ be the intersection point of $EH$ and $FG$, or…

Problem In a triangle $ABC$, suppose $AC>AB$. Let $D$ be a point on $CA$ such that $CD=AB$, $E$ be the midpoint of $AD$, $F$ be the midpoint of $BC$, and $G$ be the point where the extension of $FE$ intersects with the extension of $BA$, then $$AE=AG.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$…

Problem There are two lines that intersect at a point $Q$. Now, on one of the lines, take three points $A, \ B$, and $C$ such that $QA=AB=BC$, and on the other line, take three points $L, \ M$, and $N$ such that $LQ=QM=MN$. Then, the three lines $AL, \ BN$, and $CM$ intersect at…

Problem In a triangle $ABC$, let $AC>AB$. Let the perpendicular line from $B$ to $AC$ be $BH$. Let the perpendicular lines from a point $P$ on $BC$ to $AB$ and $AC$ be $PE$ and $PD$, respectively. Then $$PD+PE>BH.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem There are two half lines $OX$ and $OY$ starting at $O$. Let a point $P$ be within $∠XOY$ and the feet of perpendicular lines drawn from it to $OX$ and $OY$ be $Q$ and $S$, respectively. Then, if the difference between $PS$ and $PQ$ is a constant $m$, then the point $P$ is always…

Problem From a point $P$ in the given angle $∠XAY$, drop perpendicular lines $PQ$ and $PR$ to $AX$ and $AY$. If $m$ is a positive constant, then the point $P$ is on a fixed line segment such that $PQ+PR=m$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If $D$ and $E$ are the points that trisect the side $BC$ of triangle $ABC$, then $$AB+AC>AD+AE.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If the midpoint of $BC$ is $M$, and $AM$ is extended to the…

Problem From the midpoints $P$ and $Q$ of sides $AB$ and $AC$ of triangle $ABC$, draw perpendicular lines $PD$ and $QE$ to the outside of the triangle such that $$PD=\frac{1}{2} AB \qquad and \qquad QE=\frac{1}{2} AC.$$ Then, $DM$ and $EM$, which connect $D$ and $E$ to the midpoint $M$ of side $BC$, are equal and…

Problem (1) Let $M$ be the midpoint of the line segment $AB$.       Connect $M$ to a point $P$ outside this line.       If $MP<MA$, which is $∠APB$ an acute or obtuse angle?       Furthermore, what if $MP>MA$? (2) Prove that the midpoint of the hypotenuse of a right…